A188814 Sum of the "complements" of the integer partitions of n.

0, 0, 0, 1, 4, 12, 27, 57, 107, 192, 327, 538, 855, 1329, 2018, 3003, 4402, 6349, 9045, 12720, 17713, 24395, 33335, 45118, 60655, 80888, 107242, 141177, 184905, 240679, 311850, 401860, 515725, 658630, 838006, 1061561, 1340193, 1685271, 2112576, 2638727

Offset: 0

Author

Geoffrey Critzer, Apr 22 2011

Keywords

• nonn

Comments

Consider the m x k rectangle corresponding to an integer partition p of n, where m is the greatest part of p and k is the number of parts of p. Fit the Ferrers diagram of p inside its corresponding rectangle. a(n) is the number of empty spaces in all such rectangles over all partitions of n.

Examples

			a(4) = 4 because the partitions 4, 2+2, 1+1+1+1 have no empty spaces while the partitions 3+1 and 2+1+1 each have two.
		

References

• Sriram Pemmaraju and Steven Skiena, Computational Discrete Mathematics, Cambridge, 2003, page 145.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A066186, A182094, A268192.

Programs

Maple

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    elif i<1 then [0, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= n-> add(add(i, i=b(n-j, min(j, n-j)))*j, j=1..n) -n*b(n, n)[1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 22 2011, Apr 11 2012

Mathematica

f[list_]:= Total[Select[Reverse[Table[Max[list]-list[[i]],{i,1,Length[list]}]],#>0&]];
Table[Total[Map[f, IntegerPartitions[n]]],{n,0,30}]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{f, g}, If [n==0 || i==1, {1, n}, If[i<1, {0, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]}]];
a[n_] := Sum[Sum[i, {i, b[n-j, Min[j, n-j]]}]*j, {j, 1, n}]-n*b[n,n][[1]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

Formula

a(n) = Sum_{k>0} k*A268192(n,k). - Alois P. Heinz, Feb 12 2016